International Journal of Computer Vision

, Volume 106, Issue 3, pp 342–364 | Cite as

Rotation-Invariant HOG Descriptors Using Fourier Analysis in Polar and Spherical Coordinates

  • Kun Liu
  • Henrik Skibbe
  • Thorsten Schmidt
  • Thomas Blein
  • Klaus Palme
  • Thomas Brox
  • Olaf Ronneberger


The histogram of oriented gradients (HOG) is widely used for image description and proves to be very effective. In many vision problems, rotation-invariant analysis is necessary or preferred. Popular solutions are mainly based on pose normalization or learning, neglecting some intrinsic properties of rotations. This paper presents a method to build rotation-invariant HOG descriptors using Fourier analysis in polar/spherical coordinates, which are closely related to the irreducible representation of the 2D/3D rotation groups. This is achieved by considering a gradient histogram as a continuous angular signal which can be well represented by the Fourier basis (2D) or spherical harmonics (3D). As rotation-invariance is established in an analytical way, we can avoid discretization artifacts and create a continuous mapping from the image to the feature space. In the experiments, we first show that our method outperforms the state-of-the-art in a public dataset for a car detection task in aerial images. We further use the Princeton Shape Benchmark and the SHREC 2009 Generic Shape Benchmark to demonstrate the high performance of our method for similarity measures of 3D shapes. Finally, we show an application on microscopic volumetric data.


Rotation-invariance Image descriptor Fourier analysis Spherical harmonics Histogram of oriented gradients Feature design Volumetric data 



This study was supported by the Excellence Initiative of the German Federal and State Governments: BIOSS Centre for Biological Signalling Studies (EXC 294) and the Bundesministerium für Bildung und Forschung (German Federal Ministry of Education and Research) Project: New Methods in Systems Biology (SYSTEC, 0101-31P5914) – Quantitative 3D and 4D cell analysis in living organisms.

Henrik Skibbe is indebted to the Baden-Württemberg Stiftung for the financial support by the Elite Program for Post-docs. Dr. Thomas Blein was supported by a long-term post-doctoral fellowship from European Molecular Biology Organization (EMBO, ALTF250-2009). Dr. Thomas Blein and Prof. Klaus Palme are also supported by Deutsches Zentrum für Luft und Raumfahrt (DLR 50WB1022) and the European Union Framework 6 Program (AUTOSCREEN, LSHG-CT-2007-037897).


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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Kun Liu
    • 1
  • Henrik Skibbe
    • 1
    • 2
  • Thorsten Schmidt
    • 1
  • Thomas Blein
    • 3
    • 4
  • Klaus Palme
    • 3
  • Thomas Brox
    • 1
  • Olaf Ronneberger
    • 1
  1. 1.Department of Computer ScienceUniversity of FreiburgFreiburgGermany
  2. 2.Integrated Systems Biology Lab., Department of Systems ScienceKyoto UniversityKyotoJapan
  3. 3.Institute of Biology II (Botany)University of FreiburgFreiburgGermany
  4. 4.Institut Jean-Pierre BourginINRA Centre de Versailles-GrignonVersaillesFrance

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